Numerical investigations of cosmological spacetimes can be
categorized into two broad classes of calculations, distinguished
by their computational (or even philosophical) goals: 1)
geometrical and mathematical principles of cosmological models,
and 2) physical and astrophysical cosmology. In the former, the
emphasis is on the geometric framework in which astrophysical
processes occur, namely the cosmological expansion, shear, and
singularities of the many models allowed by the theory of general
relativity. In the latter, the emphasis is on the cosmological
and astrophysical processes in the real or observable Universe,
and the quest to determine the model which best describes our
Universe. The former is pure in the sense that it concerns the
fundamental nonlinear behavior of the Einstein equations and the
gravitational field. The latter is more complex, since it
addresses the composition, organization and dynamics of the
Universe from the small scales (fundamental particles and
elements) to the large (galaxies and clusters of galaxies).
However, the distinction is not always so clear, and geometric
effects in the spacetime curvature can have significant
consequences for the evolution and observation of matter
distributions.

Any comprehensive model of cosmology must therefore include
nonlinear interactions between different matter sources and
spacetime curvature. A realistic model of the Universe must also
cover large dynamical spatial and temporal scales, extreme
temperature and density distributions, and highly dynamic atomic
and molecular matter compositions. In addition, due to all the
varied physical processes of cosmological significance, one must
draw from many disciplines of physics to model curvature
anisotropies, gravitational waves, electromagnetic fields,
nucleosynthesis, particle physics, hydrodynamic fluids, etc.
These phenomena are described in terms of coupled nonlinear
partial differential equations and must be solved numerically for
general inhomogeneous spacetimes. The situation appears extremely
complex, even with current technological and computational
advances. As a result, the codes and numerical methods that have
been developed to date are designed to investigate very specific
problems with either idealized symmetries or simplifying
assumptions regarding the metric behavior, the matter
distribution/composition or the interactions among the matter
types and spacetime curvature.

It is the purpose of this article to review published
numerical cosmological calculations addressing issues from the
very early Universe to the present; from the purely geometrical
dynamics of the initial singularity to the large scale structure
of the Universe. There are three major sections: §
2
where a brief overview is presented of various defining events
ocurring throughout the history of our Universe and in the
context of the standard model; §
3
where brief summaries of early Universe and fully relativistic
cosmological calculations are presented; and §
4
which focuses on structure formation in the post-recombination
epoch and on testing cosmological models against
observations.